Certainly! Let's simplify the given mathematical expression step-by-step:
[tex]\[
\frac{3 p^2 - 35 p - 60}{4 - p}
\][/tex]
The first step to simplification is to rewrite the denominator to have the same leading term as in the numerator. Notice [tex]\(4 - p\)[/tex] can be rewritten as [tex]\(-(p - 4)\)[/tex]. Therefore, the expression becomes:
[tex]\[
\frac{3 p^2 - 35 p - 60}{-(p - 4)} = -\frac{3 p^2 - 35 p - 60}{p - 4}
\][/tex]
Now we need to simplify the expression inside the fraction. However, upon closer inspection, we find that the simplified numerator already aligns with the denominator [tex]\(p - 4\)[/tex]. Thus, rewriting and simplifying just the numerator is essential. We know:
[tex]\[
\frac{3 p^2 - 35 p - 60}{4-p} = \frac{-3 p^2 + 35 p + 60}{p - 4}
\][/tex]
Thus our final simplified expression is:
[tex]\[
\frac{-3 p^2 + 35 p + 60}{p - 4}
\][/tex]
So the completely simplified expression is:
[tex]\[
\boxed{\frac{-3 p^2 + 35 p + 60}{p - 4}}
\][/tex]
